Question: $ F = \left[\begin{array}{rr}2 & -1 \\ 1 & 0\end{array}\right]$ $ C = \left[\begin{array}{rrr}3 & -2 & 3 \\ 2 & 4 & 1\end{array}\right]$ What is $ F C$ ?
Answer: Because $ F$ has dimensions $(2\times2)$ and $ C$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ F C = \left[\begin{array}{rr}{2} & {-1} \\ {1} & {0}\end{array}\right] \left[\begin{array}{rrr}{3} & \color{#DF0030}{-2} & \color{#9D38BD}{3} \\ {2} & \color{#DF0030}{4} & \color{#9D38BD}{1}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{2}\cdot{3}+{-1}\cdot{2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{3}+{-1}\cdot{2} & ? & ? \\ {1}\cdot{3}+{0}\cdot{2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{3}+{-1}\cdot{2} & {2}\cdot\color{#DF0030}{-2}+{-1}\cdot\color{#DF0030}{4} & ? \\ {1}\cdot{3}+{0}\cdot{2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{2}\cdot{3}+{-1}\cdot{2} & {2}\cdot\color{#DF0030}{-2}+{-1}\cdot\color{#DF0030}{4} & {2}\cdot\color{#9D38BD}{3}+{-1}\cdot\color{#9D38BD}{1} \\ {1}\cdot{3}+{0}\cdot{2} & {1}\cdot\color{#DF0030}{-2}+{0}\cdot\color{#DF0030}{4} & {1}\cdot\color{#9D38BD}{3}+{0}\cdot\color{#9D38BD}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}4 & -8 & 5 \\ 3 & -2 & 3\end{array}\right] $